Here is a simple way to disable the repetitive notification of the new text messages which probably make you feel bothered. This notification is firstly introducted on iOS 4 and higher. Thus if you are searching for a way to stop it from notifying you repeatedly (or multiple beeping) each time there is a new message comes, here it's!

Open "Settings" app, like in the highlighted below:

Then tap on "Messages", scroll down to find it:

Then find the "Repeat Alert" option and then set it "Off"

Now you are freed from the Apple iPhone multiple beeping each time the incoming text message comes!